I see your dilemma!! I think a lot of functional, storage-related items can be a bit of an eyesore. I recommend using baskets, because they bring in a nice natural element and you can store things like paper towels in a cute way.
I decorated an above-fridge space like this in an episode of Mister Sister, for an example of how the baskets look! We even added in a little chalkboard for an artistic touch!
At the end of the day, the space above your fridge is usually awkward, as you said, and really isn’t somewhere you want to call a lot of attention to. A basket is a great solution because it supplies but won’t stand out too much!